What is $z$ in the Bose gas? I'm studying the ideal Bose gas, and I found this equation for the average occupation number of particles at the fundamental level:
\begin{equation}
\langle n_0 \rangle = \frac{z}{1-z}
\end{equation}
There's no degeneration due to spin, and $z$ is the fugacity defined as
\begin{equation}
z = e^{\mu/kT}
\end{equation}
But this is a problem because this makes $\langle n_0 \rangle$ negative. 
Furthermore, I see that $z$ can take values from $0$ to $\infty$, but for $z<1$ that would mean either $\mu<0$ or $T<0$, and that doesn't seem physically possible. How do you define $z$ then?
 A: $\mu $ is the chemical potential and it is negative in the case of Bose gas. 
A: Let $\mathcal{Z}$ be the grand canonical partition function of your ideal quantum gas. Since particles do not interact with each other, you can factorize the partition function:
\begin{equation}
\mathcal{Z} = \prod_{\lambda} \mathcal{z}_{\lambda}
\end{equation}
where $\mathcal{z}_{\lambda}$ is the partition function for the $\lambda$-state, written as follows:
\begin{equation}
\mathcal{z}_{\lambda} = \sum_{N_{\lambda}} \exp(-\beta(\epsilon_{\lambda}-\mu)N_{\lambda})
\end{equation}
where $\epsilon_{\lambda}$ is the energy of the $\lambda$ state, $N_{\lambda}$ its occupation number and $\mu$ the chemical potential.
The above definitions stand for both fermion and boson gases. Now, since you are working with a Bose gas there is no limit concerning the occupation number, therefore the sum over $N_{\lambda}$ will go from zero to infinity. Making a change of variable we can rewrite the partition function as
\begin{equation}
\mathcal{z}_{\lambda} = \sum_{N_{\lambda} = 0}^{\infty} r^{N_{\lambda}}=\frac{1}{1-r}
\end{equation}
which is verified only if $|r| = \exp(-\beta(\epsilon_{\lambda}-\mu)) < 1$ therefore $\mu < \epsilon_{\lambda}$
As we generally choose $\epsilon_{0} = 0$ for the ground level, the chemical potential is always negative.
Finally, the fugacity is defined as you pointed out, but for the choice of $\epsilon_0=0$ it takes values from 0 to 1.
